# Parameter Estimation of Modified Double-Diode and Triple-Diode Photovoltaic Models Based on Wild Horse Optimizer

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. PV Mathematical Modeling and Optimization Problem

#### 2.1. DDM and MDDM

_{1}, x

_{2}, x

_{3}, x

_{4}, x

_{5}, x

_{6}, x

_{7}] equivalent to [R

_{s}, R

_{sh}, I

_{ph}, I

_{s1}, I

_{s2}, ɳ

_{1}, ɳ

_{2}]. The DDM objective function is described in Equation (3). Figure 1 presents DDM detailed equivalent circuit.

_{1}, x

_{2}, x

_{3}, x

_{4}, x

_{5}, x

_{6}, x

_{7}] equivalent to [R

_{s}, R

_{sh}, I

_{ph}, I

_{s1}, I

_{s2}, ɳ

_{1}, ɳ

_{2,}R

_{sm}]. The objective function for MDDM is described in Equation (5). The equivalent circuit of MDDM is shown in Figure 2.

#### 2.2. TDM and MTDM

_{1}, x

_{2}, x

_{3}, x

_{4}, x

_{5}, x

_{6}, x

_{7}, x

_{8}, x

_{9}] is equivalent to [R

_{s}, R

_{sh}, I

_{ph}, I

_{s1}, I

_{s2}, I

_{s3}, ɳ

_{1}, ɳ

_{2}, ɳ

_{3}] where:

- -
- [I
_{ph}] Current source representing the photo-generated current; - -
- [I
_{s}_{1}] Current of the first diode representing the cell diffusion current; - -
- [I
_{s}_{2}] Current of the second diode representing the cell recombination current; - -
- [I
_{s}_{3}] Current of the third diode representing the cell grain boundaries and large leakage current; - -
- [R
_{s}] Series resistance representing the effect of the semiconductor material resistance at neutral regions; - -
- [R
_{sh}] Shunt resistance representing the effect of P-N junction leakage current resistance.

_{1}, x

_{2}, x

_{3}, x

_{4}, x

_{5}, x

_{6}, x

_{7,}x

_{8}, x

_{9}, x

_{10}] equivalent to [R

_{s}, R

_{sh}, I

_{ph}, I

_{s1}, I

_{s2}, I

_{s3}, ɳ

_{1}, ɳ

_{2,}ɳ

_{3,}R

_{sm}]. The objective function for MTDM is described in Equation (10). The equivalent circuit of MTDM is presented in Figure 4.

## 3. Wild Horse Optimizer (WHO)

- -
- Starting the problem initialization and collecting groups of horses with their leaders;
- -
- Mating and grazing of horses;
- -
- Leadership of the group (stallion);
- -
- Selecting the leaders and applying to exchange;
- -
- Obtaining the best solutions.

#### 3.1. Initialization for the Problem

#### 3.2. Grazing Behavior of Horses

#### 3.3. Horse Mating Behavior

#### 3.4. Leadership of Groups

#### 3.5. Selecting and Exchanging Leaders

## 4. Results and Discussion

#### 4.1. Results of Radiotechnique Compelec (RTC) Furnace

^{2}and temperature 33 °C [10]. The WHO parameters are crossover percentage (PC) of 0.13 and stallions’ percentage (PS) of 0.2. The four models’ accuracy was tested through the best root mean square error (RMSE) Equation (17) values obtained from the WHO. Table 1 presents the minimum and maximum range of the four models’ parameters. The best RMSE and the estimated parameters are presented in Table 2. The WHO convergence curve for all models are presented in Figure 9. The best obtained RMSE value was for MTDM followed by MDDM, and this presents the enhancement of model accuracy by the model modification. The stability of the WHO was tested by running the algorithm for 30 independent runs for MTDM and MDDM, obtaining the statistical analysis, then comparing these results with Bald Eagle Search (BES) [18] and Elephant Herd Optimization (EHO) [19], as shown in Table 3 and Table 4, and boxplot Figure 10 and Figure 11. The results of the WHO were more accurate than other algorithms for MTDM and MDDM. The authors of [20] have proposed a detailed study about parameters optimization of original and modified PV models. The authors of [20] have proposed an evaluation parameter called the polynomial equation of five degrees for the sum of squared errors (PE5DSSE) [19,20]. To compare the results of the WHO with [20], PE5DSSE was calculated and presented in Table 2. The best values of PE5DSSE proposed in [20] were 2.51 × 10

^{−5}and 2.509 × 10

^{−5}for original and modified models, respectively. The best values of PE5DSSE obtained by the WHO, which proved more accurate, were 2.50944 × 10

^{−5}and 1.52205 × 10

^{−5}for original and modified models, respectively. For detailed analysis of the obtained results by the WHO of all models, the current and power absolute errors (Equation (18)) between the real measured and the estimated current and power values were calculated and presented in Figure 12 and Figure 13 respectively. Figure 14 presents a comparison between the measured current–voltage characteristic curve and that calculated by different models. Figure 15 presents the power–voltage characteristic curve for the real PV measured data and that calculated by different models. The current–voltage and power–voltage characteristic curves for the MTDM estimated by the WHO at different temperatures are displayed in Figure 16 and Figure 17, respectively.

#### 4.2. Results of Potassium Titanate Whisker (PTW) Polycrystalline PV Panels

^{2}and temperature of 45C. The accuracy of the WHO’s estimated parameters was tested through the best RMSE and compared with recent robust algorithms in literature (I-GWO, CGO, HBO) [28,29]. A comparison between the obtained Root Mean Square Errors (RMSE) values for all compared algorithms is presented in Table 5. The convergence curves of the WHO for all compared algorithms are presented in Figure 18. The WHO achieved better speed when compared with I-GWO and HBO, as shown in Figure 18. The WHO’s stability was tested by running the algorithm for 30 independent runs and obtaining the statistical analysis of these runs, as presented in Table 6 and graphically simulated through boxplot Figure 19. For more discussion and clarification of results, the current absolute and power absolute errors for all compared algorithms are presented in Figure 20 and Figure 21. Figure 22 presents comparisons between the measured current–voltage characteristic curve and that calculated by different algorithms. Figure 23 presents the power–voltage characteristic curve for the real PV measured data calculated by different algorithms. Figure 22 and Figure 23 show the characteristic curves of the WHO, which were a better fit than other algorithms, to the real characteristic curves, a good indication about the accuracy of the PV model obtained by the WHO.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Description |

TDM | Triple-Diode Model |

DDM | Double-Diode Model |

PV | Photo Voltaic |

I | PV module output current |

V | Terminal voltage |

I_{ph} | Current source generated from photons |

ɳ_{1} | Ideality factor for the first diode (Diffusion current components) |

ɳ_{3} | Ideality Factor for the third diode (Leakage current components) |

Rs | Series resistance to represent the total semiconductor material at neutral regions resistance. |

Is1 | Current passing through the First diode |

Is2 | Current passing through the Second diode |

MTDM | Modified Triple-Diode Model |

MTDM | Modified Double-Diode Model |

SDM | Single-Diode Model |

R_{sm} | Series resistance for the losses in different regions |

RMSE | Root Mean Square Error |

Is3 | Third diode current |

ɳ_{2} | Ideality Factor for the second Diode (Recombination current components) |

R_{sh} | Shunt resistance to represent the total current leakage resistance across the P-N junction of solar cell |

K | constant of = 1.38 × 10^{−23} (J/Ko) |

q | 1.602 × 10^{−19} (C) Coulombs. |

T (Ko) | Photocell temperature (Kelvin) |

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**Figure 14.**Current–voltage characteristic curve comparison between real data and all estimated models by WHO.

**Figure 15.**Power–voltage characteristic curve comparison between real data and all estimated models by WHO.

**Figure 16.**Current–voltage characteristic curve comparison between experimental current and TDM current estimated by WHO at different temperatures.

**Figure 17.**Power–voltage characteristic curve comparison between experimental current and TDM current estimated by WHO at different temperatures.

**Figure 22.**Current–voltage characteristic curve comparison between real data and all estimated TDM by different algorithms.

**Figure 23.**Power–voltage characteristic curve comparison between real data and all estimated TDM by different algorithms.

Parameter | Solar Cell | PV Module | ||
---|---|---|---|---|

Lower Limit | Upper Limit | Lower Limit | Upper Limit | |

R_{s} | 0 | 5 | 0 | 5 |

R_{sh} | 0 | 100 | 0 | 1000 |

I_{ph} | 0 | 2 | 0 | 2 |

I_{s1} | 0 | 1 | 0 | 1 |

I_{s2} | 0 | 1 | 0 | 1 |

I_{s2} | 0 | 1 | 0 | 1 |

ɳ_{1} | 1 | 2 | 1 | 50 |

ɳ_{2} | 1 | 2 | 1 | 50 |

ɳ_{3} | 1 | 2 | 1 | 50 |

R_{sm} | 1 | 0.1 | --- | --- |

MTDM | TDM | MDDM | DDM | |
---|---|---|---|---|

R_{s} (Ω) | 0.031719 | 0.036743 | 0.0045190 | 0.036783 |

R_{sh} (Ω) | 60 | 55.50765 | 60 | 56.07530 |

I_{ph} (A) | 0.760734 | 0.760781 | 0.760347 | 0.760752 |

I_{s1} (A) | 1.97 × 10^{−}^{6} | 7.66 × 10^{−}^{7} | 2.05 × 10^{−}^{7} | 8.002 × 10^{−}^{7} |

I_{s2} (A) | 4.73 × 10^{−}^{9} | 1.00 × 10^{−}^{10} | 8.71 × 10^{−}^{7} | 2.2046 × 10^{−}^{7} |

I_{s3} (A) | 1.00 × 10^{−}^{10} | 2.26 × 10^{−}^{7} | ---- | ----- |

ɳ_{1} | 1.783574 | 2 | 1.48504 | 1.999973 |

ɳ_{2} | 1.197584 | 1.446786 | 1.579685 | 1.448974 |

ɳ_{3} | 1 | 1.446794 | ----- | ----- |

R_{sm} | 0.090031 | ------ | 0.04941 | ------ |

RMSE | 0.00076511 | 0.000982417 | 0.000783 | 0.0009832 |

PE5DSSE | 1.52205 × 10^{−}^{5} | 2.50944 × 10^{−}^{5} | 1.59406 × 10^{−}^{5} | 2.51344 × 10^{−}^{5} |

MTDM | Minimum | Average | Maximum | STD |
---|---|---|---|---|

WHO | 0.00076511 | 0.000842 | 0.000986 | 0.000125 |

BES [18] | 0.000790747 | 0.000901 | 0.00107848 | 0.00015509 |

EHO [19] | 0.001233 | 0.0059761 | 0.0131253 | 0.0039343 |

MDDM | Minimum | Average | Maximum | STD |
---|---|---|---|---|

WHO | 0.000783 | 0.000842 | 0.000960371 | 0.000103 |

BES [18] | 0.000824 | 0.001109 | 0.001603713 | 0.00043 |

EHO [19] | 0.001557 | 0.0065934 | 0.0132005 | 0.0037673 |

WHO | I-GWO | CGO | HBO | |
---|---|---|---|---|

Rs (Ω) | 1.201271011 | 1.198683773 | 1.201271 | 1.199582 |

R_{sh} (Ω) | 981.9822265 | 986.3365886 | 981.9828 | 983.629 |

I_{ph} (A) | 1.030514299 | 1.030508846 | 1.030514 | 1.030447 |

I_{sd1} (A) | 3.48 × 10^{−6} | 1.25 × 10^{−6} | 4.25 × 10^{−9} | 3.48 × 10^{−7} |

I_{sd2} (A) | 1.00 × 10^{−10} | 7.77 × 10^{−7} | 4.50 × 10^{−10} | 1.00 × 10^{−10} |

I_{sd3} (A) | 1.00 × 10^{−10} | 1.54 × 10^{−6} | 3.48 × 10^{-6} | 3.19 × 10^{−6} |

ɳ_{1} | 48.50792821 | 49.2667226 | 48.50769 | 48.60267 |

ɳ_{2} | 48.50787128 | 48.38422661 | 48.5083 | 49.52947 |

ɳ_{3} | 49.82604479 | 48.25593375 | 48.50793 | 48.56558 |

RMSE | 0.002425075 | 0.0024276291 | 0.0024251 | 0.0024281 |

Minimum | Average | Maximum | STD | |
---|---|---|---|---|

WHO | 0.002425075 | 0.002425075 | 0.002425075 | 2.16891 × 10^{−}^{16} |

I-GWO | 0.002427629 | 0.002432 | 0.002438 | 5.26003 × 10^{−} |

CGO | 0.002425075 | 0.002425092 | 0.0024251 | 1.45 × 10^{−}^{8} |

HBO | 0.0024281 | 0.002465 | 0.002528 | 5.50757 × 10^{−}^{5} |

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**MDPI and ACS Style**

Ramadan, A.; Kamel, S.; Taha, I.B.M.; Tostado-Véliz, M.
Parameter Estimation of Modified Double-Diode and Triple-Diode Photovoltaic Models Based on Wild Horse Optimizer. *Electronics* **2021**, *10*, 2308.
https://doi.org/10.3390/electronics10182308

**AMA Style**

Ramadan A, Kamel S, Taha IBM, Tostado-Véliz M.
Parameter Estimation of Modified Double-Diode and Triple-Diode Photovoltaic Models Based on Wild Horse Optimizer. *Electronics*. 2021; 10(18):2308.
https://doi.org/10.3390/electronics10182308

**Chicago/Turabian Style**

Ramadan, Abdelhady, Salah Kamel, Ibrahim B. M. Taha, and Marcos Tostado-Véliz.
2021. "Parameter Estimation of Modified Double-Diode and Triple-Diode Photovoltaic Models Based on Wild Horse Optimizer" *Electronics* 10, no. 18: 2308.
https://doi.org/10.3390/electronics10182308